Theorems · Theorem · functional analysis

ContinuousLinearMap.coprodEquivL_symm_apply

∀ {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} {G : Type u_4} (S : Type u_5) [inst : NormedField 𝕜]
  [inst_1 : Semiring S] [inst_2 : AddCommGroup E] [inst_3 : Module 𝕜 E] [inst_4 : TopologicalSpace E]
  [inst_5 : IsTopologicalAddGroup E] [inst_6 : AddCommGroup F] [inst_7 : Module 𝕜 F] [inst_8 : TopologicalSpace F]
  [inst_9 : IsTopologicalAddGroup F] [inst_10 : AddCommGroup G] [inst_11 : Module 𝕜 G] [inst_12 : TopologicalSpace G]
  [inst_13 : IsTopologicalAddGroup G] [inst_14 : ContinuousConstSMul 𝕜 G] [inst_15 : Module S G]
  [inst_16 : SMulCommClass 𝕜 S G] [inst_17 : ContinuousConstSMul S G] (f : E × F →L[𝕜] G),
  (ContinuousLinearMap.coprodEquivL S).symm f =
    (f ∘SL ContinuousLinearMap.inl 𝕜 E F, f ∘SL ContinuousLinearMap.inr 𝕜 E F)

Defined in

Mathlib.Topology.Algebra.Module.Spaces.ContinuousLinearMap

Cited by

0 results in Mathlib

Foundations

Depth 160 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedFieldSemiringAddCommGroupModuleTopologicalSpaceIsTopologicalAddGroupAddCommGroupModuleTopologicalSpaceIsTopologicalAddGroupAddCommGroupModuleTopologicalSpaceIsTopologicalAddGroupContinuousConstSMulModuleSMulCommClassContinuousConstSMul